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Line broadening caused by Coulomb carrier-carrier correlations and dynamics of
carrier capture and emission in quantum dots
Uskov, Alexander V; Magnúsdóttir, Ingibjörg; Tromborg, Bjarne; Mørk, Jesper; Lang, R.
Published in:
Applied Physics Letters
Link to article, DOI:
10.1063/1.1401778
Publication date:
2001
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Citation (APA):
Uskov, A. V., Magnúsdóttir, I., Tromborg, B., Mørk, J., & Lang, R. (2001). Line broadening caused by Coulomb
carrier-carrier correlations and dynamics of carrier capture and emission in quantum dots. Applied Physics
Letters, 79(11), 1679-1681. DOI: 10.1063/1.1401778
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APPLIED PHYSICS LETTERS
VOLUME 79, NUMBER 11
10 SEPTEMBER 2001
Line broadening caused by Coulomb carrier–carrier correlations
and dynamics of carrier capture and emission in quantum dots
A. V. Uskova)
Lebedev Physical Institute, Leninsky prospect 53, 117924, Moscow, Russia and Research Center COM,
Technical University of Denmark, Building 345v, DK-2800 Lyngby, Denmark
I. Magnusdottir, B. Tromborg, and J. Mo” rk
Research Center COM, Technical University of Denmark, Building 345v, DK-2800 Lyngby, Denmark
R. Lang
Department of Applied Physics, Tokyo University of Agriculture and Technology, Koganei-shi 184-8588,
Japan and CREST, Japan Science and Technology Corporation, Toyko, Japan
共Received 30 March 2001; accepted for publication 13 July 2001兲
Mechanisms of pure dephasing in quantum dots due to Coulomb correlations and the dynamics of
carrier capture and emission are suggested, and a phenomenological model for the dephasing is
developed. It is shown that, if the rates of these capture and emission processes are sufficiently high,
significant homogeneous line broadening of the order of several meV can result. © 2001 American
Institute of Physics. 关DOI: 10.1063/1.1401778兴
The characteristics of the optical spectra of quantum dots
共QDs兲 and, in particular, the widths of the spectral lines can
strongly affect the properties of QD based optoelectronic
devices.1 At low carrier densities, where the QD has only one
electron and one hole in the ground state, the QD luminescence spectrum is a narrow line with a width defined mainly
by the interaction of QD carriers with phonons2– 4 and by
temperature fluctuations.5 At higher carrier densities, when
the quantity of carriers inside the QD becomes larger than
one electron-hole pair, and/or carriers appear in layers surrounding the QD 关e.g. in the wetting layer 共WL兲兴, this
ground state single exciton line shifts, and new lines appear
in the QD spectrum due to Coulomb carrier–carrier
correlations.6
Coulomb interactions in QD structures can also lead to
broadening of QD spectral lines. In particular, elastic collisions of WL carriers with QD carriers can result in homogeneous line broadening of the order of several meV at very
moderate WL carrier densities.7 Matsuda et al. observed8
that, for increasing carrier density excitation 共pumping兲 of a
QD structure, broadening of the ground state exciton line
occurs in parallel with an increase of intensity of the excited
exciton line. This experimental fact can be considered as an
indication that the origin of the additional broadening is Coulomb carrier–carrier correlations inside the QD.
One should note that Coulomb correlations in QDs are
usually considered at given numbers of carriers in the QD
and neglect the continuous spectrum of allowed states above
the confined energy levels.6 Naturally, such an approach
gives only discrete energy levels, and is unable to result in
line broadening due to Coulomb interaction between the carriers. In reality, the QD is an open system, and the number of
carriers in the QD changes over time due to carrier capture
into the QD from two-dimensional 共2D兲 or three-dimensional
共3D兲 barrier states and by carrier emission from the QD into
a兲
Electronic mail: [email protected]
the barrier states. Because of Coulomb correlations inside the
QD, the changes in the electron and hole numbers in the QD
imply that the transition frequency ␻ of the basic exciton line
also changes in time, ␻ ⫽ ␻ (t). In fact, this frequency jumps
among values corresponding to the ground state single exciton, the charged exciton, the biexciton and so on. Such frequency fluctuations imply pure dephasing of the radiating
dipole of the transition considered, and can lead to homogeneous broadening of the QD spectrum,9 if these fluctuations
are sufficiently fast.
In this letter, we describe two specific mechanisms of
homogeneous broadening in a QD due to Coulomb correlations and dynamics of carrier capture and emission in QD
structures, and develop a simple model for this broadening.
In particular, we show that, if the rates of these capture and
emission processes are sufficiently high, then substantial homogeneous QD line broadening 共of the order of several
meV兲 can take place. We believe that our simple method can
be applied to a wider range of similar mechanisms.
Figure 1 illustrates processes that can lead to jumps of
the transition frequency ␻ of the considered line between two
values. In Fig. 1共a兲, one electron and one hole are in their
ground states e1 and h1 in the QD, respectively. Electron A is
captured 共for instance, due to electron–phonon interaction兲
from the barrier to state e2 in the QD, and is then emitted
back into the barrier. When electron A is in the barrier, its
wave function extends over a large volume, so the electron
charge density is negligible, and Coulomb effects of electron
A on the QD energy levels can be discarded. In this case, the
electron-hole pair radiates at the single exciton frequency
␻ 0⬘ . If electron A is at the e2 level, we have a charged exciton configuration with the radiation frequency ␻ 1⬘ of the
ground state electron-hole pair, and ␻ 1⬘ ⫽ ␻ ⬘0 due to Coulomb
interaction of the pair with electron A. Thus, because of the
carrier capture and emission processes the radiation frequency ␻ fluctuates 关 ␻ ⫽ ␻ (t) 兴 between two values, ␻ ⬘0 and
␻ 1⬘ . If n ⬘ is the population of the e2 level, the radiation
frequency is ␻ ⫽ ␻ 0⬘ at n ⬘ ⫽0 and ␻ ⫽ ␻ ⬘1 at n ⬘ ⫽1.
0003-6951/2001/79(11)/1679/3/$18.00
1679
© 2001 American Institute of Physics
Downloaded 26 Feb 2010 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp
1680
Uskov et al.
Appl. Phys. Lett., Vol. 79, No. 11, 10 September 2001
FIG. 1. Illustration of the mechanisms of line broadening. Two electron 共e1
and e2兲 and hole 共h1 and h2兲 levels are shown in the schematic QD band
diagrams. The optical transition e1–h1 with frequency ␻ (t) is considered.
共a兲 Electron A is captured into the QD e2 level, and then escapes into the
barrier. 共b兲 Hole B is captured into the h2 level with the excitation of electron A from level e1 to level e2, and is subsequently emitted into the barrier
with relaxation of electron A from level e2 to level e1.
In Fig. 1共b兲, one electron and one hole are in their
ground states. This pair radiates with frequency ␻ 0⬙ if electron A is in level e1 and hole B is located in a barrier state
共the charged exciton configuration兲. Coulomb interaction of
electron A with hole B in the barrier results in hole B being
captured into the excited h2 state and electron A being excited to the e2 state 共the Auger process兲, and we get the two
exciton configuration. This capture-excitation process is
shown in Fig. 1共b兲 by thin arrows. The transition frequency
of the considered radiating pair 共e1–h1兲 in this configuration
is ␻ 1⬙ , and is different from ␻ 0⬙ . Coulomb interaction of hole
B and electron A inside the QD can lead to emission of hole
B from the QD and relaxation of electron A back to level e1
共the processes are shown by bold arrows兲. We see again that
due to the capture-excitation and emission-relaxation processes the radiation frequency ␻ ⫽ ␻ (t) fluctuates between
␻ 0⬙ and ␻ ⬙1 . If n ⬙ is the population of levels e2 and h2, i.e.,
the population of the QD with excited excitons e2–h2, then
the radiation frequency is ␻ ⫽ ␻ 0⬙ at n ⬙ ⫽0 and ␻ ⫽ ␻ 1⬙ at
n ⬙ ⫽1.
Broadening of the QD spectrum due to the mechanisms
in Fig. 1 can be described within the framework of the stochastic equation,
ẋ⫽⫺ ␥ x⫺i ␻ 共 t 兲 x⫹ f 共 t 兲 ,
共1兲
for the complex amplitude x(t) of the radiating dipole. The
transition frequency ␻ ⫽ ␻ (t) randomly fluctuates between
␻ 0 and ␻ 1 关␻ 0 ⫽ ␻ 0⬘ , ␻ 1 ⫽ ␻ ⬘1 for the mechanism in Fig. 1共a兲
and ␻ 0 ⫽ ␻ ⬙0 , ␻ 1 ⫽ ␻ 1⬙ for the mechanism in Fig. 1共b兲兴. Spontaneous emission leads to decay of the oscillator with the rate
␥, and to noise fluctuations described by the Langevin force
f (t). The force is assumed to be delta correlated, i.e.,
具 f (t) f * (t⫹ ␶ ) 典 ⫽C ␦ ( ␶ ), where the angled brackets 具 典 indicate an average over fluctuations. The coefficient C is
related to the rate ␥ and the intensity of the spectral line.9
The
S( ␻ )
spectrum
is
expressed
as
S( ␻ )
⫹⬁
dte ⫺i ␻ t 具 x(t)x * (t⫹ ␶ ) 典 . In the calculation of S( ␻ ) we
⬀ 兰 ⫺⬁
follow the method in Ref. 10. It is easy to show that
S 共 ␻ 兲 ⫽C/4␲ ␥ 关 具 g̃ 共 ␻ 兲 典 ⫹ 具 g̃ 共 ␻ 兲 典 * 兴 ,
共2兲
where 具 g̃(z) 典 is the Laplace transform of the average Green
izt
function 具 g(t) 典 for Eq. 共1兲: 具 g̃(z) 典 ⫽ 兰 ⫹⬁
0 dte 具 g(t) 典 . Thus,
the problem of calculating S( ␻ ) is reduced to finding
具 g̃(z) 典 .
We consider the process ␻ (t) as a Markov process with
two discrete values or, in different terminology, a Kubo–
Anderson process 共KAP兲 with two values.10 It means the
following. If P 0 (t) is the probability that ␻ (t)⫽ ␻ 0 关n⫽0, n
is the population n ⬘ of level e2 for the mechanism in Fig.
1共a兲, and n is the population n ⬙ of excited excitons 共e2–h2兲
for the mechanism in Fig. 1共b兲兴, and if P 1 (t) is the probability that ␻ (t)⫽ ␻ 1 (n⫽1), then
Ṗ 0 共 t 兲 ⫽⫺ v c P 0 ⫹ v e P 1 ; Ṗ 1 共 t 兲 ⫽⫹ v c P 0 ⫺ v e P 1 ,
共3兲
where v c ⫽1/␶ c ( v e ⫽1/␶ e ) is the capture 共emission兲 rate for
electron A in Fig. 1共a兲 or the capture-excitation 共emissionrelaxation兲 rate for excitons in Fig. 1共b兲, and ␶ c ( ␶ e ) is the
respective capture 共emission兲 time. Equations 共3兲 have the
stationary solution P s0 ⫽1⫺n̄, P s1 ⫽n̄, where n̄⫽ v c /( v c
⫹ v e ) is the average population. One can show that, in this
stationary regime, the correlation 具 ␦ ␻ (t) ␦ ␻ (t⫹ ␶ ) 典 for the
deviation ␦ ␻ ⫽ ␻ (t)⫺ 具 ␻ 典 is given by 具 ␦ ␻ (t) ␦ ␻ (t⫹ ␶ ) 典
⫽ 具 ␦ ␻ 2 典 e ⫺ v 兩 ␶ 兩 where
v ⫽ v e / 共 1⫺n̄ 兲 .
共4兲
10
The average Green’s function for this KAP is
具 g̃ 共 z 兲 典 KAP⫽
g̃ sKAP共 z⫹i v 兲
,
1⫺ v g̃ sKAP共 z⫹i v 兲
共5兲
where g̃ sKAP(z) is the so-called static Green’s function,
g̃ sKAP共 z 兲 ⫽
1⫺n̄
n̄
⫹
.
␥ ⫹i 共 ␻ 0 ⫺z 兲 ␥ ⫹i 共 ␻ 1 ⫺z 兲
共6兲
Substitution of Eqs. 共5兲 and 共6兲 into Eq. 共2兲 gives the S( ␻ )
spectrum.
We see that the S( ␻ ) spectrum depends on the average
population n̄. If n̄⫽0 (n̄⫽1) the spectrum is Lorentzian,
centered at ␻ ⫽ ␻ 0 ( ␻ ⫽ ␻ 1 ) with a full width at half maximum 共FWHM兲 equal to 2␥, i.e., there is no additional line
broadening at n̄⫽0 and 1. For 0⬍n̄⬍1, the S( ␻ ) spectrum
can have a more complicated shape, which depends on n̄ and
the relation among ␥, v and ⌬ ␻ ⫽ 兩 ␻ 1 ⫺ ␻ 0 兩 . If v Ⰶ ␥ ,
S共 ␻ 兲⬀
1⫺n̄
n̄
⫹
⫹cc.
␥ ⫹i 共 ␻ 0 ⫺ ␻ 兲 ␥ ⫹i 共 ␻ 1 ⫺ ␻ 兲
共7兲
i.e., the spectrum is the sum of two unbroadened Lorentzians
at ␻ ⫽ ␻ 0 and ␻ ⫽ ␻ 1 with weighting of 1⫺n̄ and n̄, respectively. This is the so-called static case, and the condition v
Ⰶ ␥ means simply that the frequency ␻ (t) does not change
its value 共␻ 0 or ␻ 1 兲 during radiative decay of the optical
dipole, which takes place on a time scale of ⬃1/␥.
Consider the more interesting case of v Ⰷ ␥ . Figure 2
shows the normalized S( ␻ )/S max spectra for different n̄ and
v e . In all these plots, we assume that ␻ 0 ⫺ ␻ 1 ⫽3 meV,
which is in accordance with the calculations in Ref. 5, and
␥ ⫽0.06 meV. Figure 2 shows that the shape of the S( ␻ )
spectrum depends strongly on n̄ and on the relation between
v e and ⌬␻. At n̄⫽0, we have a line at ␻ 0 with width of 2␥.
If v e ⫽0.3 meV 关Fig. 2共a兲兴 共␶ e ⬃3 ps, v e Ⰶ⌬ ␻ 兲, an increase
of n̄ results in broadening of the line at ␻ ⫽ ␻ 0 and the ap-
Downloaded 26 Feb 2010 to 192.38.67.112. Redistribution subject to AIP license or copyright; see http://apl.aip.org/apl/copyright.jsp
Uskov et al.
Appl. Phys. Lett., Vol. 79, No. 11, 10 September 2001
FIG. 2. Normalized S( ␻ )/S max spectra. In all plots, ␻ 0 ⫽0, ␻ 1 ⫽⫺3 meV,
and ␥ ⫽0.06 meV. Thick solid line: n̄⫽0; dashed line: n̄⫽0.3; dot-dashed
line: n̄⫽0.5; dotted line: n̄⫽0.7; thin solid line: n̄⫽0.99. v e ⫽共a兲 0.3 and
共b兲 2 meV.
pearance of a broad line at ␻ ⫽ ␻ 1 . At n̄⫽0.5, we observe
two resolved peaks 关dot-dashed line in Fig. 2共a兲兴. With a
further increase of n̄, the line at ␻ ⫽ ␻ 0 disappears, and the
line at ␻ ⫽ ␻ 1 grows and narrows to a width of 2␥ 关thin solid
line in Fig. 2共a兲兴. The case of v e Ⰶ⌬ ␻ is thus similar to the
static case, Eq. 共7兲, but is accompanied by substantial line
broadening of the order of v e .
Increasing v e means that the two broadened lines come
close to each other, and finally merge. At v e ⭓0.5⌬ ␻
⫽1.5 meV 共corresponding to ␶ e ⬃1 ps兲, only one line is observed, and the line shifts from ␻ 0 to ␻ 1 with increasing n̄.
Figure 2共b兲 ( v e ⫽2 meV) illustrates this behavior. When the
average population n̄ increases from 0 to 0.5, the line broadens, reaching a maximum width of ⬃1.5 meV at n̄⫽0.5. A
further increase of n̄ implies narrowing of the line. One can
show that, at v e ⭓⌬ ␻ , the line at n̄⫽0.5 is Lorentzian with
linewidth of ⬃⌬ ␻ 2 / v .
Thus, the spectral line shape and its width depend quite
strongly on the parameter v , which characterizes the population dynamics in the QD, and can be very different from
those in the static case 关Eq. 共7兲兴. For v e ⭓0.5⌬ ␻ , one instead
of two lines is observed. For the mechanism shown in Fig.
1共a兲, one can say that the single exciton and the charged
1681
exciton merge into one common exciton. Thus, it appears
impossible to consider these two excitons separately if the
dynamics of carrier capture and emission are fast.
As discussed above, only one line will appear in the QD
spectrum, if the emission rate v e is higher than approximately 0.5⌬ ␻ ( ␶ e ⬍2/⌬ ␻ ). In general, the difference frequency ⌬␻ between two excitons depends on the QD size
and is of the order of several meV.5 It implies that for observation of one merged or composite line, instead of two wellseparated exciton lines, the emission time must be shorter
than approximately 1 ps. In the mechanism illustrated in Fig.
1共a兲, such short times may result due to electron–
longitudinal optical 共LO兲 phonon interaction in single phonon capture and emission processes.11 It implies that the perturbing e2 level must be relatively shallow, with its bound
energy less than one LO phonon energy below the continuum
band edge. On the other hand, by considering the mechanism
in Fig. 1共b兲, emission times of the order of ⬃100 fs can be
reached,11 and the e2 level can be relatively deep 共in large
QDs兲. Thus, in large QDs the situation with one broad line
instead of two well-resolved lines can be realized with the
mechanism of Fig. 1共b兲, rather than with that of Fig. 1共a兲. In
this case, for ⌬ ␻ ⬃10 meV the linewidth will be of the order
of ⬃⌬ ␻ 2 / v e ⬃10 meV. Note also that the linewidth depends
strongly on n̄, which for this mechanism is the population of
the QD with excited excitons 共e2–h2兲. Since the population
also determines the intensity of the luminescence of this excited exciton, broadening of the ground state exciton 共e1–h1兲
occurs concurrently with the increase of the luminescence
intensity. This agrees with the observation in Ref. 8 where it
was shown that the width of the ground state exciton line
starts to increase with pumping from a certain level, and that
this additional broadening is correlated with the intensity of
the luminescence of the excited exciton line.
In conclusion, the dynamics of carrier capture and emission in quantum dots can significantly modify the optical line
shape due to Coulomb interactions and can lead to strong
line broadening 共of several meV兲.
The work of one of the authors 共A.V.U.兲 was supported
by the Russian Federal Program ‘‘Integration’’ 共Project No.
A0155兲, by RFBR 共Project No. 01-02-17330兲, by the Danish
Research Council within the framework of program
‘‘SCOOP,’’ and by the Otto Mo” nsted Foundation.
1
D. Bimberg, M. Grundmann, and N. N. Ledentsov, Quantum Dot Heterostructures 共Wiley, New York, 1999兲.
2
T. Takagahara, Phys. Rev. B 60, 2638 共1999兲.
3
X.-Q. Li and Y. Arakawa, Phys. Rev. B 60, 1915 共1999兲.
4
A. V. Uskov, A.-P. Jauho, B. Tromborg, J. Mo” rk, and R. Lang, Phys. Rev.
Lett. 85, 1516 共2000兲.
5
M. Arzberger and M.-C. Amann, Phys. Rev. B 62, 11029 共2000兲.
6
See, for instance, U. Hohenester and E. Molinary, Phys. Status Solidi B
201, 19 共2000兲, and references therein.
7
A. V. Uskov, K. Nishi, and R. Lang, Appl. Phys. Lett. 74, 3081 共1999兲.
8
K. Matsuda, T. Saiki, H. Saito, and K. Nishi, Appl. Phys. Lett. 76, 73
共2000兲.
9
M. Sargent, III, M. O. Scaully, and W. E. Lamb, Jr., Laser Physics
共Addison–Wesley, New York, 1974兲.
10
A. Brissaud and U. Frisch, J. Math. Phys. 15, 524 共1974兲.
11
R. Ferreira and G. Bastard, Appl. Phys. Lett. 74, 2818 共1999兲.
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